Isomorphism and equivariance of the Whittaker-space embedding

Let $\mathcal{O}\subset\mathscr{X}_{Q,n}$ be a $W$-orbit, let $I(\chi)$ be the relevant principal-series representation, and let $\operatorname{Wh}_\psi(I(\chi))_\mathcal{O}$ and $\operatorname{Wh}_\psi(I(\chi))_\mathcal{O}^{\sharp}$ denote the corresponding Whittaker subspaces. Let $T(w,\chi)$ and $T(w,\chi)^{\sharp}$ be the homomorphisms induced by the intertwining operator $T(w,\chi)$, and let $\iota(\chi)_{\psi,\mathcal{O}}$ be the natural embedding.

Whittaker-space isomorphism conjecture. For every $W$-orbit $\mathcal{O}\subset\mathscr{X}_{Q,n}$, the embedding is an isomorphism

$$\iota(\chi)_{\psi,\mathcal{O}}:\operatorname{Wh}_\psi(I(\chi))_\mathcal{O}\simeq\operatorname{Wh}_\psi(I(\chi))_\mathcal{O}^{\sharp}.$$

Moreover, it is equivariant in the sense that

$$\iota({}^w\chi)_{\psi,\mathcal{O}}\circ T(w,\chi)_{\psi,\mathcal{O}}=T(w,\chi)_{\psi,\mathcal{O}}^{\sharp}\circ\iota(\chi)_{\psi,\mathcal{O}}.$$

This identifies the two descriptions of the relevant Whittaker spaces and makes them compatible with intertwining operators. The supplied text gives no resolution status beyond the assertion itself.